To convert the logarithmic equation $\ln x = 5$ into an equivalent exponential equation, we use the property of natural logarithms. The equation $\ln x = 5$ means that $x$ is the number such that the natural logarithm (base $e$) of $x$ is 5.

The conversion rule is: $\ln x = y \Rightarrow x = e^y$

So, applying this to $\ln x = 5$: $x = e^5$

Answer:

The equivalent exponential equation is: $x = e^5$

Would you like a more detailed explanation or have any questions?

Here are five related questions to expand your understanding:

1. How do we convert a general logarithmic equation with different bases to an exponential form?
2. What is the approximate value of $e^5$?
3. How do we solve for $x$ in equations of the form $e^{x} = k$?
4. What are some applications of natural logarithms in calculus?
5. How can we use logarithmic properties to simplify expressions like $\ln(a \cdot b)$?

Tip: Remember that $\ln x = 5$ can be rewritten in terms of exponentiation with base $e$, the base of natural logarithms, making conversions straightforward.